Could you take the sprocket from a 21 speed rear wheel on a bike and put it in the rear wheel of a 18speed bike?

First off, give up this 18/21 speed nonsense. Tha may be how many dealers label their bikes, but if you gonna mix n' match you're better off referring to the number of rear sprockets. 18 speed can be both 2 front and 9 rear(2x9) or 3 front 6 rear(3x6), with your 21-speed is a 3x7 Anyhow, a 3x6 is limited to a design called a freewheel, where the mechanism that allows you to coast is integrated with the sprockets A 3x7 can be either a freewheel or a freehub(where sprockets and coasting mechanism are treated as separate pieces).

And yoy can't put sprockets from a freehub wheel onto a freewheel wheel If both the 3x6 and the 3x7 are freewheels then you can move the freewheel (= the sprocket pack) from one over to the other. But the 7-speed freewheel takes up more axle space than the 6-speed, so you'l likely have to add some spacers I'll post a link where there are nice pics and explanations on how to tell them apart.

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